ESSENTIALS OF HYDRAULIC TURBINE ANALYSIS AND DESIGN
ESSENTIALS OF HYDRAULIC TURBINE ANALYSIS AND DESIGN
Hydraulic turbines extract energy from the gravitational potential of water sources or from the kinetic energy of flowing water or from a combination of the two. These turbines are generally classified as either impulse or reaction. Reaction turbines are further classified as radial and mixed-flow (Francis) turbines or as axial-flow or propeller turbines.
Efficiency generally governs which turbine type is selected. Figure 16.14[1] plots efficiency against specific speed (Ns) for the three turbine types.
[pic]
Where h is in feet and ne is that rpm corresponding to optimum operating efficiency.
IMPULSE TURBINES (See Figure 16.1 and 16.3, Finnemore)[2]
Impulse turbines operate under relatively high heads and low flow rates. One or more nozzles convert available energy into kinetic energy, most of which is transferred to buckets attached to a rotating wheel (runner). The resulting shaft torque drives a generator or other machinery. Windage, fluid friction, turbulence, separation and leakage cause the principal losses.
• Nozzle Design
V0 Vi1
Figure 1
The ideal exit velocity, Vi1, is calculated from the Bernoulli equation:
[pic]
The ideal velocity is multiplied by a velocity coefficient, Cv , to account for friction and turbulence. Cv varies from about 0.95 (needle valve partly closed) to 0.99 (needle valve fully opened).[3]
[pic]
The actual quantity rate is obtained by multiplying the ideal rate by a discharge coefficient, Cd. The discharge coefficient is the product of the velocity coefficient and the contraction coefficient, Cc, (ratio of area of emerging jet to the area of the nozzle at the discharge point). The value of Cc is about 0.94.[4]
Conservation of mass leads to:
[pic]
Where : [pic]
• Nozzle Dimensions
The nozzle diameter at discharge is made about 20% greater than the calculated diameter of the jet. The nozzle should terminate in a cone of 30-45(.[5]
• Rotational Velocity
Calculate rpm from the specific speed that results in reasonable efficiency. Guidelines follow:
Head (ft) Specific Speed (ns)
1000. 5.0 – 5.5
2000. 4.0 – 5.0
[pic]
Where: n = rpm; [pic] = shaft horsepower; H = turbine head, ft. If the turbine drives a generator select a rotational velocity equal to the nearest synchronous speed calculated from:
[pic]
Where: f = frequency (60 cycles per second in U.S.)
p = number of poles
• Runner Diameter (Dp). The runner diameter is determined from a formula similar to that for the centrifugal pump.[6]
• [pic]
Enter Figure 13[7] with ns to obtain an estimate of[pic], a factor based upon experience.
• Absolute Bucket Entering Velocity
[pic]
Where: U = peripheral velocity of a point on the pitch diameter of the bucket. Because V2 must be greater than zero, however, U is decreased somewhat.
Let [pic]be the angle through which the water is turned relative to the bucket. For maximum work,[pic]; however, to prevent water from striking the succeeding bucket, it must be somewhat less, say, 150-1600.
• Bucket Shape and Dimensions[8]
The bucket shape on either side of the vertical centerline is semi-ellipsoidal. A sharp-edged “splitter” divides the flow, one-half going to either side.
Approximations for bucket dimensions follow:
Width - B = 3d
Depth - D = 0.85d
Length -L = 2.6d
Where: d = jet diameter at rated capacity.
• Power Analysis (See Figure 16.4, Finnemore)[9]
[pic] [pic] [pic]
[pic]
Where: [pic] tangential velocity
U = peripheral velocity
[pic]mass flow rate
[pic]power[pic]
REACTION (FRANCIS) TURBINES
(See Figures 16.8 and 16.11, Finnemore)
The Francis turbine consists of a runner with shrouded buckets, somewhat analogous to a centrifugal pump. Wicket gates that direct the flow and control the power and speed surround the runner. The water enters the turbine through a spiral scroll casing with a changing area to keep the entering velocity constant. The usual range for available head is 75-1600 feet; for specific speed, 15-100.
• Design
The turbine buckets are tangent to the entering relative velocity at the tip. They are designed to leave without appreciable tangential velocity (whirl). Thus, the exit term in Euler’s equation can be neglected, and the angle between the exiting absolute velocity and the tangent,[pic], can be set at 90(. Refer to Figure 2 for the velocity triangles. The power equation becomes:
• [pic]
[pic]
• Selection of Speed
Economics calls for high rotational speeds resulting in small units. Considerations of efficiency, cavitation and structural strength, however, place an upper practical limit on speed.
Figure 16.14 (see page 1) plots efficiency against specific speed (Ns) for the three principal turbine types. Figure 16.16 (following page) plots specific speed against maximum effective head (h). To select a practical speed, enter Figure 16.16 with maximum effective head and draft head and select the highest specific speed outside of the cavitation region. Then calculate the resulting rpm. Enter Figure 16.14 to estimate efficiency. Select the nearest synchronous speed corresponding to the value calculated from specific speed.
• Selection of Runner Diameter
The runner diameter is determined from a formula similar to that for the centrifugal pump.[10]
[pic]
Where D is in inches and the peripheral-velocity factor, [pic], is found from Figure 16.14.
The number of buckets can be estimated from:[11]
[pic]
The usual range is 21 for low and 12 for high specific speed. Refer to Marks’ for other runner dimensions.
• Draft Tubes
After passing through the turbine, the water enters a draft tube, Figure 16.11, (page 6). The purpose of this tube, which is an integral part of the turbine design, is threefold:
1) To permit the turbine to be set above the tailwater level without loss of head.
2) To recover a reasonable amount of the kinetic energy leaving the runner by
diffuser action.
3) To facilitate inspection and maintenance.
Note that the pressure at the upper end of the draft tube is below atmospheric thus limiting the height above the tailwater because of cavitation considerations. The velocity at the upstream end of the tube ranges from 24 to 30 ft/s; at the lower end, 5-7 ft/s. The included angle of the diffuser tube should be kept reasonably small, say 8-12(, to limit losses due to separation. Typical loss coefficients are:[12]
Cone Angle Loss Coefficient
8( 0.23
12( 0.33
[pic]
Where: K = Loss Coefficient and numerals 1 and 2 refer to entering and leaving stations.
-----------------------
[1] Finnemore, E.J. and Franzini, J.B., Fluid Mechanics with Engineering Applications, 10 ed., p. 707, McGraw Hill, 2002.
[2] Id., p. 686.
[3] Id., p. 695.
[4] Id., Figure 11.13, p. 506.
[5] Marks’ Standard Handbook for Mechanical Engineers, 8 ed,, p. 9-145, McGraw-Hill, 1978.
[6] See class notes, “Centrifugal Pump Design,” equation (13), p. 5.
[7] Marks’ p. 9-144.
[8] Id., Figure 14, p. 9-145.
[9] Finnemore, p. 688.
[10] See class notes, “Centrifugal Pump Design,” equation (13), p. 5.
[11] Marks’ , p. 9-141.
[12] Finnemore, Figure 8.20, p. 310.
-----------------------
V2=Vr2
Vr1
W1
W2
V1
U
U
Figure 2
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